# E. T. Jaynes' subjectivism vs measurement of distributions

In his paper, E. T. Jaynes argues that entropy is a measure of our ignorance about a system. As such, the probability distribution of states $$\{p_k\}$$ has to be chosen in the most unbiased way, thus maximizing the entropy constrained to all the available information. This is a subjectivist point of view because treats probabilities as description of our ignorance, rather than as an intrinsic property of the system. He also claims that the reason statistical mechanics works is that the distributions are sharply peaked and, as long as the peak is at the correct position, its shape is not that relevant.

With the development of computers and experiments, however, now we are able to simulate distributions of states in a system or measure actual equilibrium fluctuations at a high resolution (with optical tweezers, for example). Going beyond macroscopic quantities, thus, we can simulate/measure actual probability distributions of states. Measurements show that these are indeed the distributions that maximize entropy (at constant temperature, for instance, it's the Boltzmann distribution). How would a subjectivist argue then that the probability of states are due to our lack of information about the system? If I can measure those distributions, they look very objective to me.

Going beyond macroscopic quantities, thus, we can simulate/measure actual probability distributions of states.

This is a misunderstanding. One never measures probability, the verb does not apply to the noun. In such simulations/calculations one may record some numbers, such as number of times the system was found in some region of phase space (or number of times system assumed some definite microstate). Such numbers can be divided by total number of observations or total number of time points, but this only gives frequency of occurences in that simulation, an artefact that depends on initial condition that may not repeat itself with different initial condition. It can serve as estimate of the probability, but itself is not the probability, which is supposed to abstract from details such as the initial condition. Jaynes provides a coherent way to think about the probability and a way to find probabilities in a number of cases of interest in statistical physics, using the maximum information entropy principle. Of course, one should test, if possible, usefulness of so determined probabilities, for example through computer simulations of concrete cases.

• I agree that all you can measure is frequency of occurences. At equilibrium, however, these frequencies converge to well defined values as you increase the number of measurements, independently of the initial conditions. I'm biased because I was trained in the objectivist spirit, but I'm still having hard time to see why it's obvious to measure frequencies that are precisely the same as the "working probabilities" "guessed" via the maximum entropy principle. – Botond Feb 15 '19 at 22:26
• The match between observed frequencies and maxent probabilities isn't obvious, and it isn't always the case. However, it just is the most probable thing to happen, if all available knowledge was taken into consideration when predicting the probabilities. It is like with the law of large numbers: there is no guarantee that statistics on large number of experiments will show agreement with the probability derivation, but if the derivation is right, it is very probable. – Ján Lalinský Feb 15 '19 at 23:14
• You answer completely ignores the problem of which interpretation of probability is adopted. For a pure frequentist, probability are defined by the frequency of the measurements! – GiorgioP Feb 16 '19 at 5:44
• @GiorgioP I do not think that adoption is a problem that needs to be addressed to answer the question, as bayesian/jaynesian probability is well established in physics and elsewhere. – Ján Lalinský Feb 16 '19 at 12:46
• I have a different feeling. Frequentist approach to probability has played an important role in ensemble theory in Statistical Mechanics and in Statistical interpretation of QM and in part that point of view is still present. In any case, it is a fact that the usual training of physicists tends to ignore the problem of probability interpretation, notwithstanding its central role in many subjects. – GiorgioP Feb 16 '19 at 14:12

On the basis of the present classification of probability interpretations, I would not classify Jaynes' approach as subjectivism, even if Jaynes presented his approach in this way. Certainly it is a Bayesian point of view, but not a subjectivist one. Both Bayesian and subjectivism look at probabilities as a measurement of the state of knowledge. But such a measurement, for a subjectivist, is directly related to the quantification of a personal belief, while objective Bayesian approach tries to build "objective" priors, for example using maxent method, where the role of the objective knowledge about the system is quite clear.

In a Bayesian approach to probability (and then to Statistical Mechanics) the success of the theory should be seen as a confirmation about the choice of the priors. Therefore, there is no conflict between the fact that one can check the consistence between measured frequencies and probabilities and the knowledge about the system which has been used to establish the priors.